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Sobolev W0 basics question

  1. Oct 5, 2008 #1
    As I understand it, one way to define Sobolev spaces is to say they are the collection of functions with weak derivatives up to some order, and this space is called [itex]W^{k,p}[/itex].

    On the other hand, some stuff I am reading now defines it differently. Here they define the Sobolev space [itex]W_0^{k,p}[/itex] as the completion of [itex]C_0^\infty[/itex] (the space of smooth functions with compact support) with respect to the Sobolev norm.

    Now, by the definition elements of [itex]W_0^{k,p}[/itex] are equivalence classes of Cauchy sequences in [itex]C_0^\infty[/itex]. How do we know that these equivalence classes actually represent functions? For example, what does it even mean to "integrate" an equivalence class?
  2. jcsd
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